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Aspects of Chaos

James A. Yorke likens chaos to the pachyderm in the tale of the blind men and the elephant. In that story, a handful of blind men come upon an elephant, and each makes contact with a different part of the animal. The man who handles the ear is reminded of a fan; the one who hugs the leg thinks of a pillar; he who fingers the tail concludes that the creature is ropelike. Each man’s limited scope prevents him from grasping the big picture, and this is where Yorke sees a chaos parallel.

“If you only understand one aspect of chaos,” the University of Maryland professor told his audience at the MAA Carriage House on November 17, “then you’re really limited in what you understand about chaos.”

Yorke considers the ability to frame definitions one of the “great powers of math,” but he hesitated to constrain his presentation by precisely defining “chaos.” He instead characterized chaotic behavior in contrast to the other two possibilities—steady state and periodic/quasiperiodic—for bounded dynamical behavior. What sets chaotic systems apart, Yorke explained, is that “nearby trajectories diverge from each other very fast.”

This sort of rapid divergence need not get out of hand, however. It can be controlled, as Yorke illustrated with the example of NASA’s International Sun-Earth Explorer 3 (ISEE 3). Part of this spacecraft’s mission was to intercept the tail of Comet Giacobini-Zinner. To power the craft across millions of miles of space on a meager 200 pounds of fuel, however, it was necessary for ISEE 3 to make a series of lunar swing-bys, worked out by NASA scientist Robert Farquhar and his team. ISEE 3 gained energy on each pass behind the moon, and lost energy on each pass in front.

James Yorke presents "Aspects of Chaos" as part of MAA's Distinguished Lecture Series.

“Repeated near passes of the moon basically act like a chaotic phenomenon,” Yorke said, noting that if ISEE missed one pass by even a mile or two, its entire trajectory would be “way off.” Scientists monitored the spacecraft’s movement, though, and used additional rocket burns to make course corrections. Yorke attributes the success of the ISEE 3 mission to “controlling chaos.”

Yorke followed the dizzyingly loopy path of ISEE with another arresting visual. In it, marbleized-looking swirls of red, blue, and chartreuse conveyed information about the correspondence between how a forced damped pendulum moves—it can flip, clockwise or counterclockwise, or just swing back and forth—and various combinations of initial angle and angular velocity.

It turns out, Yorke explained, zooming in on the graphic, that every point on the boundary between two different colored regions is actually a triple boundary point. Wherever two colors come together, all three do. That the regions, called Wada basins, are “so intermingled,” Yorke said, means that, “if you start on the boundary, you have chaotic behavior.” You also see “chaos being revealed in amazing pictures.”

Chaos scholarship can, however, be as practical as it is picturesque. Yorke alluded to work he has done in chaos-based weather prediction and on determining when a carrier of HIV is most infectious. As a result of research conducted by Yorke and colleagues, scientists can better forecast weather events on Mars, and public health officials can argue more convincingly for the implementation of broad-based HIV screenings.

Yorke ended his tour of things chaotic with another animal metaphor. He recalled a question posed by nineteenth-century mathematician and physicistJames Clerk Maxwell: If we knew everything about a leopard—every atom—could we then predict how the leopard would behave? No, Maxwell contended, because of the chaotic (he wouldn’t have used that word) nature of atomic collisions.

But Maxwell also recognized, Yorke said, that “the leopard will not change his spots,” because the chaos is in some sense bounded. Within the body of a leopard—as within a population susceptible to HIV—the immense number of interactions yields, thanks to the “law of large numbers,” a deterministic system. “Chaos implies statistical regularity,” Yorke summarized.